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# Test the improper integral ∫ e-x/2 for convergence

Test the following improper integral for convergence:

The integral converges.

Proof. We compute directly,

# Test the improper integral ∫ 1 / (x3 + 1)1/2 for convergence

Test the following improper integral for convergence:

The integral converges.

Proof. We know the integral converges (example #1 on page 417 of Apostol). Applying the limit comparison test (by the note to Theorem 10.25 on page 418, which says that if then the convergence of implies the convergence of .) we have

Since we know converges the theorem establishes the convergence of

# Test the improper integral ∫ e-x2 for convergence

Test the following improper integral for convergence:

The integral converges.

Proof. We have

The first integral converges since it is a proper integral. For the second integral, since we have . But, we know converges (by Example #4 on page 418 of Apostol with ). Hence, we have established the convergence of

# Test the improper integral ∫ x / (x4 + 1)1/2 for convergence

Test the following improper integral for convergence:

This integral diverges.

Proof. By example 1 (page 417) of Apostol, we know the integral diverges. We then apply the limit comparison test (Theorem 10.25 on page 418),

Thus, by the limit comparison test, we have established the divergence of